String cobordism refers to the Thom spectrum for the 7-connected cover of BO, the classifying space for real vector bundles. I will describe progress toward a description of its 3-primary homotopy type in joint work with Vitaly Lorman and Carl McTague. It supports a map to tmf (the spectrum associated with topological modular forms) which is surjective in homotopy groups.
This is joint work with Wolfgang Pitsch. We illustrate how equivariant stable homotopy methods can help us recognize the structure of a conjugation space, as introduced by Hausmann, Holm, and Puppe. We first explain their definition and present a characterization in terms of purity (obtained in previous joint work with Nicolas Ricka). We then perform equivariantly Floyd's construction from the 1970s of a pair of 5- and 10-dimensional manifolds with four cells, relying on Lück and Uribe’s work on equivariant bundles. The 10-dimensional one is a conjugation space.
Derivator theory, initiated by Grothendieck and Heller in the '90s to correct the shortcomings of triangulated categories, motivated a lot of research regarding the foundation of (∞,1)-category theory, and its applications to algebraic geometry/topology.
For a 2-category theorist, a (pre)derivator is a familiar object - (a suitably co/complete) prestack on the category cat of small categories - and yet still little is known about the formal properties of the 2-category PDer. The present talk is motivated by the belief that time is ripe for a more conceptual look into the foundations of derivator theory, and that far from being a mere exercise in style, such a conceptualization yields many practical advantages.
After briefly outlining the essentials of "formal category theory'' (2-categories can be used to organize the theory of "categories with structure" just as category theory organizes the theory of "sets with structure"), I will report on a conjecture regarding the possibility to provide a "yoneda structure" or a "proarrow equipment" to the 2-category of pre/derivators. Under suitable assumptions, these are equivalent ways to equip PDer with a calculus of Kan extensions, and building on prior work of Di Liberti and myself, this allows to speak about "locally presentable" and "accessible" objects (showing that Adamek-Rosický and Renaudin's definitions eventually coincide); the overall goal is to provide a suitable form of special/general adjoint functor theorem for a morphism of prederivators (such a theorem would simplify a lot the life of the average algebraic geometer).
How does the dimension of the first cohomology grow in a tower of covering spaces? After a tour of examples of behaviours for low-dimensional spaces, I will focus on arithmetic manifolds. Specifically, for towers of complex hyperbolic manifolds, I will describe how to bound the rates of growth using results from Langlands functoriality.
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, 𝒜t(G) of finite injective dimension, a homology theory π∗𝒜t taking values in 𝒜t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence
Ext𝒜t(G)∗ , ∗ (π∗𝒜t(X), π∗𝒜t(Y)) → [X,Y]∗G
for rational G-spectra X and Y.
This approach should be viewed as an analogue of the Cousin complex in algebraic geometry. It is expected that a similar method will apply to other tensor triangulated categories with finite-dimensional Noetherian Balmer spectra.
The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called 'h-principle results' and try to explain some of the ideas behind the ones I am most familiar with.
Mackey functors play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk, I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of constant Mackey functors for G=ℤ/2. Our decomposition leads to a computation of the Balmer spectrum of the derived category. We extend these results to classify all finite modules over the equivariant Eilenberg-MacLane spectrum Hℤ/2.
Factorization systems (both weak and strong) are commonly defined as consisting of two classes of maps satisfying a certain orthogonality relation and a factorization axiom. The standard definition of algebraic weak factorization system, involving comonads and monads, is rather different. The goal of this talk will be to describe an equivalent definition of algebraic weak factorization system emphasising orthogonality and factorization.
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of ∞-categories, and exploit this to prove many classical theorems about monads in the ∞-categorical setting. Among other things, we prove that the category of algebras for an accessible monad on a locally presentable ∞ category is locally presentable. We also apply the result to construct examples of ∞-categorical monads from pretheories.
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